\documentclass[a4paper,10pt]{article}
\usepackage{graphicx}
\usepackage[width=18cm,right=1.5cm,height=28cm,top=1cm]{geometry}
\usepackage{asymptote}
\usepackage{xepersian}
\settextfont[Scale=.9]{Times New Roman}
\setlatintextfont[Scale=1.5]{Times New Roman}
\setdigitfont[Scale=1]{Persian Modern}
\setmathfont[Scale=1.5]{Times New Roman}
\begin{document}
\begin{LTR}
problem 1:

%
Let $ABC$ be a triangle with center $O$ and orthocenter $H$. choose points $P,Q,R$ on perpendicular bisectors of $BC,CA,AB$ such that $AQ=AR,BR=BP,CQ=CP$. Let $K,S$ be orthocenter and circumcenter of $\triangle PQR$ respectively. Prove that $OS\parallel HK$.
\end{LTR}

%
\begin{figure}[h]
\begin{asy}
import graph; size(12cm); 
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ real xmin = -20, xmax =36, ymin =-24, ymax =30.5; /* image dimensions */
pen uuuuuu = rgb(0.26666666666666666,0.26666666666666666,0.26666666666666666);
/* draw figures */
draw(circle((13.33760695751346,-0.7146186348428829), 4.577807876421848)); 
draw(circle((9.41075177125982,-0.9247843409032798), 11.606922204372264)); 
draw((1.9995957973239615,8.008043329456468)--(-1.1774302842963,-5.679891470795235)); 
draw((-1.1774302842963,-5.679891470795235)--(20.76483396166046,-3.334236700253186)); 
draw((20.76483396166046,-3.334236700253186)--(1.9995957973239615,8.008043329456468)); 
draw((12.221206429950058,3.7249731048879298)--(9.41075177125982,-0.9247843409032813)); 
draw((9.41075177125982,-0.9247843409032813)--(0.41108275651383075,1.1640759293306164)); 
draw((9.41075177125982,-0.9247843409032813)--(9.79370183868208,-4.507064085524211)); 
draw((12.221206429950058,3.7249731048879298)--(8.760181438816288,-0.7737842854337396)); 
draw((9.682993561696748,-3.4714512160031803)--(8.760181438816288,-0.7737842854337396)); 
draw((9.682993561696748,-3.4714512160031803)--(12.221206429950058,3.7249731048879298)); 
draw((13.33760695751346,-0.7146186348428829)--(9.41075177125982,-0.9247843409032813), linewidth(1.2) + red); 
draw((3.989167515436178,0.9089748731367748)--(2.765495932168487,0.8434838402146063), linewidth(1.2) + red); 
/* dots and labels */
dot((1.9995957973239615,8.008043329456468),linewidth(3.pt) + blue); 
label("$A$", (1.7161178220302067,8.341705701892046), NE * labelscalefactor,blue); 
dot((-1.1774302842963,-5.679891470795235),linewidth(3.pt) + blue); 
label("$B$", (-1.6636442897704045,-6.087278698487454), NE * labelscalefactor,blue); 
dot((20.76483396166046,-3.334236700253186),linewidth(3.pt) + blue); 
label("$C$", (21.05225682704332,-3.552457114637001), NE * labelscalefactor,blue); 
dot((9.41075177125982,-0.9247843409032813),linewidth(3.pt) + uuuuuu); 
label("$O$", (9.613061987102789,-1.4726035073750914), NE * labelscalefactor,uuuuuu); 
dot((12.221206429950058,3.7249731048879298),linewidth(3.pt)); 
label("$Q$", (12.147883570953246,4.052007636914357), NE * labelscalefactor); 
dot((8.760181438816288,-0.7737842854337396),linewidth(3.pt) + uuuuuu); 
label("$R$", (8.280655769950624,-0.49767212897107094), NE * labelscalefactor,uuuuuu); 
dot((9.682993561696748,-3.4714512160031803),linewidth(3.pt) + uuuuuu); 
label("$P$", (9.158094010514244,-3.7474433903178053), NE * labelscalefactor,uuuuuu); 
dot((3.989167515436178,0.9089748731367748),linewidth(3.pt) + uuuuuu); 
label("$K$", (4.1209485554267955,1.0947157890888288), NE * labelscalefactor,uuuuuu); 
dot((13.33760695751346,-0.7146186348428829),linewidth(3.pt) + uuuuuu); 
label("$S$", (13.480289788105411,-0.5301698415845383), NE * labelscalefactor,uuuuuu); 
dot((2.765495932168487,0.8434838402146063),linewidth(3.pt) + uuuuuu); 
label("$H$", (2.2685789364591527,1.0622180764753615), NE * labelscalefactor,uuuuuu); 
dot((9.41075177125982,-0.9247843409032798),linewidth(3.pt) + uuuuuu); 
dot((0.41108275651383075,1.1640759293306164),linewidth(3.pt) + uuuuuu); 
dot((9.79370183868208,-4.507064085524211),linewidth(3.pt) + uuuuuu); 
dot((11.382214879492212,2.336903314601641),linewidth(3.pt) + uuuuuu); 
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); 
/* end of picture */
\end{asy}
\end{figure}

%
\begin{LTR}
problem 2

%
Let $ABC$ be a triangle and $P$ be an arbitrary point. Let $Q$ be the inverse of $P$ WRT $\odot(\triangle ABC)$. Let $M$ be the midpoint of $PQ$ and $S,T$ be the projections of $Q$ on $AC,AB$ respectively. prove that $\angle SMT=\angle APB$.
\end{LTR}

%
\begin{figure}[h]
\begin{asy}
import graph; size(7cm); 
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ 
pen dotstyle = black; /* point style */ 
real xmin = -17.83382969905937, xmax = 30.580959440654926, ymin = -9.906339462577114, ymax = 12.487719631381971; /* image dimensions */

/* draw figures */
draw(circle((7.,0.8985381668107605), 7.057433728857513)); 
draw((2.5309179713991963,6.3606510051593945)--(0.,0.)); 
draw((14.,0.)--(-5.213631323204422,0.)); 
draw((14.,0.)--(-3.8467282272389665,9.897636929818244)); 
draw((-5.213631323204422,7.432932653469829)--(-3.8467282272389665,9.897636929818244)); 
draw((-5.213631323204422,7.432932653469829)--(-5.213631323204422,0.)); 
draw((-5.213631323204422,7.432932653469829)--(7.,0.8985381668107605)); 
draw((-0.6920696683930831,5.013859507536967)--(-3.8467282272389665,9.897636929818244)); 
draw((-0.6920696683930831,5.013859507536967)--(-5.213631323204422,0.)); 
draw((2.5309179713991963,6.3606510051593945)--(3.8294919864182555,2.594786361604105)); 
draw((3.8294919864182555,2.594786361604105)--(0.,0.)); 
/* dots and labels */
dot((2.5309179713991963,6.3606510051593945),dotstyle); 
label("$A$", (2.4298772274108895,6.8828645240603885), NE * labelscalefactor); 
dot((0.,0.),dotstyle); 
label("$B$", (-0.35986965858626735,-0.4719227208412364), NE * labelscalefactor); 
dot((14.,0.),dotstyle); 
label("$C$", (14.425788837198665,-0.3197547088777545), NE * labelscalefactor); 
dot((3.8294919864182555,2.594786361604105),dotstyle); 
label("$P$", (4.027641353027443,2.9518575483371063), NE * labelscalefactor); 
dot((7.,0.8985381668107605),linewidth(3.pt) + dotstyle); 
label("$O$", (7.22316960426055,1.0243960634663356), NE * labelscalefactor); 
dot((-5.213631323204422,7.432932653469829),linewidth(3.pt) + dotstyle); 
label("$Q$", (-5.7871954186171,7.618343248550551), NE * labelscalefactor); 
dot((-0.6920696683930831,5.013859507536967),linewidth(3.pt) + dotstyle); 
label("$M$", (-0.7656510238222175,4.321369656008443), NE * labelscalefactor); 
dot((-5.213631323204422,0.),linewidth(3.pt) + dotstyle); 
label("$T$", (-5.330691382726656,-0.5733680621502243), NE * labelscalefactor); 
dot((-3.8467282272389665,9.897636929818244),linewidth(3.pt) + dotstyle); 
label("$S$", (-3.7075659217828556,10.281283457911483), NE * labelscalefactor); 
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); 
/* end of picture */
\end{asy}
\end{figure}

\end{document}